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Oops!

In the past,  I firmly stated that you should teach what you can teach a particular student or a group of students, at a given time. I believed that this was the case with teaching Robert about percents.  I analyzed possible difficulties, I planned ahead, I chose (Or so I thought.) appropriate curriculum and yet,  a small error, tiny omission lead me to a standstill and Robert to unlearning.

A few days ago, I was  preparing  new folders from Take It to your Seat series to practice with Robert previously taught skills.  As I looked through the pages of the Math workbook, I came across a section on finding the  prices of the items on sale.  That was not a topic we had previously addressed but the skill seemed useful and relatively easy to teach. It required the application of  two simple operations: subtraction and multiplication by decimal. When I, however, analyzed the possible teaching methods I couldn’t decide if it would be better to first subtract percent from 100% and then multiply, or first multiply by proper decimal and follow with subtraction.  Moreover, I was not sure if in some cases it wouldn’t be more useful to change percent into a regular fraction and replace the multiplication with  the division by a reciprocal of that fraction.  I was not sure which approach would be the easiest to apply in real life situation.

As  I looked for the support or/and inspiration in  the Unit 2 devoted to percents in Momentum Math  level G, I decided it would  make sense to skip the Take It  to Your Seat for now and introduce percents to Robert in a systematic way.  Momentum Math appeared to be a good tool for that.  I looked through lessons A to H, with the  exception of  the Lesson E, as its title suggested that it was the lesson only about  common percents: 25% and 50%.  Judging by the title, I concluded that it would be useful and easy to digest chapter. There was no reason, to look at it more carefully.

So we went: Lesson A, Lesson B, Lesson C.  No major problems.  We skipped Lesson D.  It was about comparing data, which could be done later, if at all.   We begun Lesson E.  The unpleasant surprise awaited me on the second page.   Beside the problems I was prepared to work with Robert, such as

25% of 60 =?

there were also problems I was not ready to teach Robert at this moment:

25% of  ? = 30

Had I noticed that sooner, I would have practiced or reviewed prerequisite skills or skip this lesson replacing it with my own worksheets. But I did not notice.

Oops!

Robert doesn’t take lightly switching from one kind of problem to another.  He doesn’t like to feel lost. He despise being confused.

More importantly, he HATES when he cannot complete all the worksheets prepared for him for a given day.

As soon as I tried to explain to Robert that this lesson should be left for another day, Robert put both arms on the page to prevent it from being taken away.

I tried to lead Robert through the second sort of problems, but the only thing I achieved was to confuse him. As a result, he started making errors in the first category of problems.

I suggested that we skip just those few difficult problems, but Robert persistently tapped on them letting me know that this was not an option.

I cornered myself.

So I cheated.  I asked Robert to bring me a glass of water from the kitchen.  As soon as he left, I hid the three of the five pages. This scheme, unfortunately, was not a bullet proof.  In the past, for different reasons, I did the same thing only to have Robert looking indefatigably for the missing worksheets and almost always finding them. Still, there was a small chance, he would give up..

When he returned, he started searching.

After a few minutes, he found the missing pages in the binder with  worksheets prepared for the future dates. He hesitated for a second, glimpsed at me,  and…  closed the binder without taking the worksheets out.

I could not believe! How was that possible?

Did he decide that the problems were too puzzling?  Was his distaste for being in a state of confusion stronger than his need to complete the unpleasant task and thus let him conquer his obsessive compulsive behavior?

Was his effort to pretend, that he did not find those worksheets a sign that he had that evasive thing…Theory of Mind?

Or am I assuming too much?

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by krymarh on January 28, 2013  •  Permalink
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Posted by krymarh on January 28, 2013

https://krymarh.com/2013/01/28/oops/

On Doubts and Multiplication

I wanted to write a simple text about teaching multiplying large numbers, but as it is the case with all the teaching, many questions immediately popped out. Consequently, what supposed to be a logical, clear, linear chain of thoughts , became an entangled knot of half doubts. Those doubts happened to be as, if not more, important than the multiplication itself. Yet, to appreciate the importance of  doubts I have to talk about issues Robert and I encountered while teaching/learning multiplication.

Robert mastered times table in the first half of 2006.

He did not have any problem with multiplication of two or three digit number by a one digit when there was no need to regroup.  For instance 52 times 3.

When Robert was learning to multiply two digit numbers by one digit with regrouping I asked him to always write the result of first multiplication (ones by ones) on the side and then rewrite it in a split way.  The ones under ones but tens above tens.  For instance when counting   43 times 4 , Robert first writes 12 on the side and then he rewrites it by placing 2 under 3 and 1 above 4.  And so on.

Of course there are other multiplication algorithms and some of them are much more intuitive, but I chose this one as it is the most frequently used.

I have to say, that it did help a lot that Robert became good at mental addition finding the sums  similar to those: 17+4, 56+8  as this skill is utilized a lot during multiplication.

Well, not exactly.  When Robert started multiplying larger numbers, he was not able to do all the required additions mentally.  I often asked him to write on the side, and he dutifully followed.  But as he continued to do that I often stopped him just after he wrote the numbers under each other.  I let him look at them , but didn’t let him write the answer below.  I wanted him to say the sum just by looking at addenda and do regrouping in his mind. I think that was an important step to develop ability to add in his mind.  It was during multiplying when Robert mastered mental addition (in the narrow range required to find products).

When Robert started multiplying by two digit numbers I used (as many curricula suggest) vertical and horizontal lines to help organize partial products in proper spaces.

Robert learned to multiply three and four digit factors  by two digit factors.  He made , however, mistakes of forgetting to multiply by ten’s digit.  The horizontal lines let him remember.  Still, it was a problem I addressed by making worksheets in which I mixed up multiplying by one digit with multiplying by two digits. Interspersing problems this way demanded that Robert pays attention.  This problem presented itself  many times, specially after periods of time  when we did not count products.

Multiplying by three digit factor, is still a problem.   Robert can be completely confused and often cannot find even partial product.  When, around 2008 I dealt with this problem for the first time, I tried to hide under tiny stickers two out of the three digits, so Robert concentrated on only one.  (That is what we did initially while multiplying by two digit number).  But even so, it was still very baffling.  Robert  wanted to multiply the hundred’s digit only by hundred’s digit.  Since it was so hard for Robert to follow with those numbers and for me to find a way to make it easier, I decided to stop working intensively on that skill. But over the next four years, while Robert was learning other things, I kept returning  to the task.  I wanted Robert to learn, but mainly I wanted to understand HOW HE LEARNS.

Clearly, the hundred’s digit was a problem. Somehow Robert was not able to connect it with one’s and ten’s digit of the other factor.

I drew three arrows from hundred’s digit to all the digits of the factor above, as the way of visualizing the process.  Each arrow had a tiny number attached to it: 1, 2,3.   That seemed to help a little.  But not much.

Today, I wrote just 4 multiplication problems.  Two on each piece of paper.  With vertical lines and horizontal lines. I asked Robert to cross out each digit in the bottom number he had already multiplied by.  With some prompting, he kept crossing one’s, then ten’s digit until he got to hundred’s.  He was still a little uneasy about connecting hundred’s with one’s and ten’s of the other factor, but did not make mistake.  Crossing off seemed to make a difference.  But we are not of the woods yet.

Now I pose the question, which probably any person reading this blog would like to ask , “Why am I teaching Robert multiplication he probably will not have a chance to use in any of the adult programs he might be included in the future? ”   Why don’t I just give him simple one digit problems (If I really need to do something with him.) he can solve easily and feel good about himself?  That is after all,  what Robert’s school is doing almost every day this year.

First , Robert doesn’t feel proud when he fills easy worksheets.  To the contrary, he does feel very proud in those magical moments when he grasps something new, something he could not do before. 

Second, had Robert had language and was able to communicate with me sufficiently, I might find other things more appropriate to teach. But when Robert learns multiplication, he learns to follow a complicated algorithm, he learns to follow steps, to pay attention, to redirect his attention as he moves from one digit to another. 

Third, the fact that Robert had so many problems with multiplying by three digit numbers was a REASON to teach that skill, as a way of maybe (just maybe) addressing the deficit which caused those problems in the first place. 

Ability to multiply is not a goal in itself.  I am teaching multiplication for the same reason I am trying to teach everything else. To discover and possibly address  the learning (thinking?) difficulties  Robert has.  I don’t know any better method of understanding  Robert and helping him to understand the world. 

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by krymarh on December 28, 2012  •  Permalink
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Posted by krymarh on December 28, 2012

https://krymarh.com/2012/12/28/on-multiplying-and-on-doubts/

What About Reminder?

Robert memorized multiplication facts during his 2005/2006 year in Collaborative program.  I  cannot describe how Robert learned multiplication facts because my role was only marginal.  It, probably, helped  that in 2004, Robert knew already how to count by five, two, and ten or that I practiced with him counting by  other numbers.   I made worksheets to practice changing repetitive addition into multiplication and  worksheets  in which a new multiplication fact was scattered  among the facts that Robert already mastered.  The Collaborative Program, however, led the way.  I think that the worksheets the teachers used relied a lot on patterns. I recall a worksheet that introduced multiplying nine by consecutive numbers by emphasizing a pattern of results:  increasing ten’s digit and simultaneously decreasing one’s digit. I am not sure  if at that time Robert and I worked on those lessons from Saxon Math that presented multiplication as an array of objects.  Maybe not yet.  I recall vividly, that Robert learned quickly and almost effortlessly as it was always the case when the school assumed responsibility for teaching.  Unfortunately, not much later, Robert was forced to leave this program.

For the next four months I was teaching Robert at home much more intensively than before.  We practiced multiplication facts and family of facts  as a way to almost mechanically tie multiplication to division.  (That is what we had done before when Robert had been learning to memorize subtraction facts). That allowed Robert to memorize division facts relatively quickly.  The problem arose when reminders had to be taken into account. It is one thing to remember that 32 :8=4 and another to remember what is 34, 35, or 38  divided by 8.

We started by learning to  divide by 2.  On top of each page, I wrote  the multiples of two on a number line in large print and the odd numbers in small.  Robert had to place each dividend on the number line and then look to the left to find the closest multiple.  For 17, he pointed to 16, for 9, he found 8 and so on.  In the next step, we skipped number line, but Robert still had to write all the multiples of two with empty spaces between them and  locate the odd number somewhere between the multiples.  For 11, for instance, he had to point to empty space between 10 and 12,   then move his finger to 10, and proceed from here.
Finally, when Robert practiced division by two, he was prompted to help himself by listing the multiples of 2 in order but only when he was unsure or confused.

We repeated these steps for all the divisors from 3 to 9.  Of course while learning to divide by 3 with a reminder, Robert also practiced, on a separate page, dividing by 2.  Then I mixed two kinds of problems on the same page.

I have to say, however, that we didn’t go  past dividing by more than 12.

Learning to divide with a reminder helps with long division and with changing improper fractions to mixed numbers.  It is clear that  it would be beneficial  to practice dividing by divisors larger than 12.  That, however, is still a problem, and when we have to do that, I just ask Robert to find a few multiples of the divisor (by multiplying).  Cumbersome process that still requires hand over hand (prompt over prompt) guidance.  We do that only rarely, when Robert wants to finish the problem which is a little over his (or mine) head.

The algorithm of dividing by numbers larger than 12 depends on  ability to round numbers up or down.  This is a very hard skill to teach, and Robert doesn’t have it.

For school this is NOT a functional skills, so instead Robert is practicing counting values of nickles: 5cents, 10 cents, 15 cents…

Didn’t he do that 10 years ago?

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by krymarh on December 8, 2012  •  Permalink
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Posted by krymarh on December 8, 2012

https://krymarh.com/2012/12/08/what-about-reminder/

Counting Errors. His and Mine

Robert was making many mistakes today (November 6) as he was trying to solve word problems that required adding and subtracting fractions. The word “more” confused him again when it was a part of a phrase, “How much more?”, so he wanted to add the quantities that should be subtracted. Although he found common denominators without problems, he kept forgetting the whole numbers when they were part of the mixed fractions.  He couldn’t understand the way he had to regroup the fractions and change one whole into a fraction of a given denominator so it could be added to the fraction’s part of the mixed number.  He made too many mistakes for me to address at the same time.

The best solution would be to put the packet of 5 worksheets away and return to it after practicing separately or in a  chain of related problems of increasing difficulties all the skills required for successful completion of those worksheets.

I couldn’t do that, because when Robert starts working on a packet of worksheets, he has to finish working on the worksheets. So I led Robert through each and every problem from the beginning to end.  That didn’t make any educational sense.  Robert was not learning anything about solving word problems that required adding and subtracting fractions.  Even worse, I was teaching him not to trust himself, be passive, and helplessly wait for others to solve problems for him.

Yes, Robert was making many errors but his errors were results of a giant blunder I had committed by asking him to work on this folder.

I miscalculated, to put it gently.

I should have known that asking  Robert to solve the problem that had two steps would lead to errors even if he could fluently address each step separately. I was asking him to put together three skills, each very different from the other: choosing the proper operation (a sum or a difference), finding common denominators, not forgetting the whole numbers in the process, and changing one whole number into a fraction.

I cannot even say, that before I started working with Robert on this unit,  I was not aware of difficulties he (and I)  might encounter.  Almost one year ago, I worked with Robert on the same packet, just for diagnostic purposes.  I wanted to know how far he could go on his own; how difficult would it be for him to follow all the steps. I found out that was very hard.

Consequently, a year ago, I designed a pretty good, step by step program to help Robert learn.

1. Robert worked on worksheets that only had problems of the format 1-2/3. Subtracting a fraction from 1.

2.Robert worked on worksheets on which each problem of the form 1-2/3 was followed by a few problems of the form 2-2/3 or 5-2/3

3.Robert worked on problems in which he had to change mixed fraction into the improper fraction: 2 and 3/4= 11/4

4. Robert worked on problems where he had to change just 1 out of the whole number into the fraction: 3 and 1/4= 2and 5/4

5.  I made worksheets which required Robert to do the three steps in a row: 1-2/3 followed by (4-2/3),then by (4 and 1/3 – 2/3), and  finally by (4and 1/3- 2and 2/3)

I am using the word “and” as well as the parenthesis in the above expressions only for the purpose of this writing. 

This is exactly what I should have done today. Instead of rushing to complete one more chapter in the book on fraction I should have planned to practice those, well, prerequisite skills over a few more days or weeks.

I know Robert  well enough, to predict that  my errors would lead to his.

.

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by krymarh on November 9, 2012  •  Permalink
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Posted by krymarh on November 9, 2012

https://krymarh.com/2012/11/09/counting-errors-his-and-mine/

Segment by Segment to Infinity

I, too, am asking myself , “What is the point of teaching Robert elements of basic geometry? What is the point of introducing these abstract concept of point, line, ray, or line segment?  Is it not enough that Robert names geometrical shapes in two and three dimensions?

What is the point of teaching that a line segment has two endpoint, a ray has one, and a line doesn’t have any ends but extends itself to infinity in both directions?

For the last few days Robert has been bringing  from school pages of  simple  operations on coins. A dime and a nickel. A change from a dollar. The  activities on those worksheets seem so realistic, normal, needed….functional for a student with severe developmental delays and almost no language.

But… Robert has an ATM card where his SSI payments are deposited every month.  He uses his ATM to withdraw money or pay for his McDonald meals, once a week grocery shopping, or movie tickets.   From time to time, he writes checks under my supervision and directions.  He doesn’t have yet access to banking on-line as  I am afraid that Robert could be enticed into transferring money into someone’s else account.

No, I don’t think there is anything functional in teaching Robert counting coins.

On the other hand, those abstract concepts of points, rays, line segments, and lines might one day calm his unspoken restlessness as he will  look, from whatever place he finds himself to be,  on the starry sky above. He will see the stars like endpoints of the  line segments that extend themselves into infinite universe.  With those few concepts from Euclidean geometry he might have a spark of understanding and for a moment he will find himself to be a  part of the universe.

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by krymarh on November 5, 2012  •  Permalink
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Posted by krymarh on November 5, 2012

https://krymarh.com/2012/11/05/segment-by-segment-to-infinity/

Teaching While Learning How to Teach

In one of the previous posts I wrote about three steps in teaching Robert a skill.  The first step is the hardest to explain.  I don’t expect Robert to learn the skill.  I lead him step by step through the procedure, I talk to him knowing that he doesn’t understand most of what I am saying, no matter how simple they seem to me.  It is as if I asked him to look through the window of a moving train and notice an object we were passing by.  I wouldn’t expect Robert to notice any of the specific features of that object.  For now a realization that there is something out there will suffice. We will study it intensively another time.  That would be a second step. Next we would proceed to finding the same object in its different manifestations and various environments… In other words Robert  would generalize the skill.

When I described those steps  I omitted the most important function of the first step. Besides exposing Robert to an existence of something new, this step is for me to learn how to teach Robert. I observe Robert to know  which of the steps I am leading him through he can climb on his own, which words he recognizes, which he doesn’t, and what other support I will need to provide to assure learning.

There were times when  I skipped this first step.  I jumped into intensive teaching without strategic reconnaissance.  I expected Robert to learn something I didn’t know how to teach.  I was putting pressure on Robert not on myself.

What I am writing is not about a teacher being prepared  for a lesson (for instance by writing a lesson plan).  The lesson plan is an important tool, but for children like Robert it is just  like a hypothesis.  It doesn’t assure learning.  It only sets the stage for testing of the hypothesis.  The first step, as I understand it, is about checking the hypothesis. You don’t know how to teach before you start to teach and analyze responses – good and bad, complete and partial. The teacher has to learn from those responses and then develop  proper strategies, check them again, revise them again until he or she finds the best approach to teaching this specific skill to this specific student.

Just yesterday I tried to teach Robert to divide fractions. I had a lesson plan.  I had already practiced with Robert prerequisite skills like finding reciprocals.   I used simple words.  The words he knew – multiply , divide, flip the fraction over, reciprocal, reciprocal..

He couldn’t grasp the division.  That should be fine. The mistake was that I expected him to learn and consequently subjected him to the same method over and over.  That lead to more errors and repeated failures.  The failures lead to frustration and learned helplessness…

I shouldn’t pressure Robert  to learn during that phase.  I should be the one who was supposed to learn during that time. I had to take a breath and think about this failed attempt.

When I did,  I noticed  that although Robert understands when I say, “Multiply four by seven.  Divide 42 by six”  he doesn’t understand me when I say, “multiply instead of dividing”.  I observed that for Robert finding reciprocal to the separately standing fraction is not the same as finding reciprocal during division. I came to a conclusion that relying on language concepts that Robert used only in limited numbers of applications was not working. I understood that I would have to to experiment with different methods of presenting the new information.

In the end  I used abstract algebraic formula: “a/b : m/n = a/b * n/m .  This approach I would use with “typical” students. It seemed much too abstract for  Robert.  But since  I was only experimenting it was no harm in trying it. I introduced this formula not expecting Robert to learn it and use it.  I pointed it to him as if that was that object we had  seen from the passing train.  Somehow this formula seemed for Robert easy to follow.  Much easier than my verbal directions. I found a way to teach.  The phase one was over. Let’s move to the next.

One might point out to me, that even during a failed phase of intensive teaching I  was still learning how to teach.  That is true.  Yet I was also subjecting Robert and myself to unnecessary frustration by rigidly sticking to one approach instead of investigating its effects and being ready to flexibly adjust it.  I should have taken a breath, led Robert through the activity (just stopping it without finishing would leave negative residues which also should be avoided) as calmly as possible.  Then, I should have taken time and rethink the whole process in connection with everything I knew about Robert and started over.

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by krymarh on April 17, 2012  •  Permalink
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Posted by krymarh on April 17, 2012

https://krymarh.com/2012/04/17/teaching-while-learning/

Looking for Variables

Robert learned numbers (symbols,counting, names, order) by the time he was five years old. Soon after that he learned to add 1 to any other number.

And then for the  next five years he DID NOT LEARN  one math fact.

For five years! Nothing!

Not because he was not taught.  He was taught at school, he was taught at home.

School used not only flash cards, this antediluvian staple of American math education, but also cleverly designed program that involved cute counters and number cards.

I used counters, number line, two math curricula from SRA and yes, in my desperation, against my better judgement, and betraying my principles, I tried the  flash cards too.

Nothing worked!

Susan, the clinical supervisor of Robert’s program, sadly advised me to accept that Robert would never learn to add numbers.

Such outcome seemed both unavoidable and unacceptable.

The same week or month a parent on old ME-List advised another parent to use Saxon Math with her child.  Since the price was not prohibitive I ordered it without really knowing what I was buying.

Eureka!

The order in which  math facts were introduced seemed  counter-intuitive to me.  The first math facts to remember were additions of duplicates: 1+1, 2+2, 3+3 and so on.  It took Robert a week to add duplicates until 5+5.  It took him another week to memorize additions from 6+6 to 10+10.

Although it seemed so strange at first I quickly understood how much simpler  8+8 was than 2+4.  In the first addition there was  one number to remember.  So it sufficed to just learn that 8 was related to 16.  In the second addition there were two numbers.  You had to remember them both and that was hard for Robert. Which of the two numbers  is the  important one?  The first one or the last one?   It cannot be 4 because 1+4 is not the same as 2+4. It cannot be 2 because…

The second step in memorizing addition fact was to practice adding double plus one in the form: 1+2, 3+4, 7+8…

The problem 7+8 was written next to 7+7.  Since Robert knew the first fact and knew how to add one to any number he didn’t have much problem with 7+8.

Yet, that was just a mechanical approach.  Without seeing 7+7 first Robert was not able to solve 7+8.

I added new worksheets.  I wrote 7+8 first and 7+7 next so Robert would learn to use the second problem  as a support for the first one. Just looking to the right  was an important step.  Next, I wrote only 7+8 and next to it I drew empty squares.  Robert filled those squares with supporting addition 7+7 =14 and then solved  7+8.  In the  next step  I wrote just 7+8  and let Robert write himself   the next column.  Finally I wrote 7+8  but when Robert wanted to write 7+7 next to it I blocked the space.  He had to write an answer to 7+8 as he deduced it in his head without writing 7+7 on the paper.

So Robert knew how much was 8+9 before he knew 2+4.

Similar trick I used with practicing adding 2 to the number.

Someone (surprisingly a parent) asked me why I spent so much time on teaching addition instead of just introducing a calculator.

The answer is complicated. Of course it is nice that Robert can add, subtract, multiply and divide large numbers.  Typical people and peers with disabilities  who because of Robert  terrible problems with communication tend to  dismiss him after first encounter, might  realize that he has some relatively advanced skills.

I also believe that in this process Robert learned not just how much is 7+8 but also some strategies that he might one day apply to solving other problems.

However,  the main point of teaching Robert math facts was to LEARN how ROBERT LEARNS.  To find out what works, what doesn’t. During this process  I realized that Robert had problem with short and/or working memory. But I also discovered  that Robert learns through patterns.  Moreover, I  found out that even when he doesn’t have any visual support Robert  can still solve problems by using his mind.

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by krymarh on March 31, 2012  •  Permalink
Posted in Math, Teaching Out of Autism
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Posted by krymarh on March 31, 2012

https://krymarh.com/2012/03/31/looking-for-variables/

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